044501-5 K.A.Strain and B.N.Shapiro Rev.Sci.Instrum.83,044501(2012) The estimator equation in (8)receives the control input i that minimizes this cost function in the chosen interval is opti- and the stage 1 sensor signals y in the Euler frame and esti- mal.For x,R is a scalar because it is a decoupled DOF by the mates the corresponding modal signals g using the modeled symmetry of the pendulum.Coupled DOFs such as y,pitch, state space matrices Am,Bm,Cm.The subscript m indicates and roll must be considered simultaneously resulting in a ma- the matrices refer to the modal frame.The goal is now to de- trix. sign the estimator feedback gain L. The two competing performance criteria of damping time The observer separation principle allows us to keep the and technical noise are stated above.The sensor noise is a already established control scheme,provided that the estima- known measured quantity approximately 7x 10-11 m/Hz tor has an accurate model of the pendulum,which is the as- beyond 10Hz.Since the contribution to mirror motion drops sumption here.Note that the estimator replaces the direct sig- off quickly in the frequency domain,we will only consider nal transformation in (5). the contribution at 10Hz,the start of the gravitational wave The estimator is designed using the Linear Quadratic detection band. Regulator(LQR)technique which solves the cost function in These two performance criteria are represented in the op- Egs.(9)and(10): timization routine with the cost function: =( Jg(R)=max(T2)+max(N2). (12) (9) DOF DOF with R=argmin(J&). (13) with Lm=argmin(). (10) R Here,T,is the normalized stage 4 settling time;chosen The Q matrix weights the accuracy of the modal state es- as the maximum settling time of all the DOFs normalized by timation while the R matrix weights the cost of using a noisy the goal,10s.Similarly,Nis the maximum sensor noise con- measurement.Here 3 is defined as tribution to the mirror relative to the respective requirement at 10 Hz for each DOF considered. (11) As an example,the results of Eg.(12)are plotted in Figure 4 for the x DOF where R is scalar valued.The opti- where Lm is the estimator feedback matrix determined mal value occurs at R=0.06.Since the total cost is less than by(10). 1,both design requirements are met at this optimal point.The An estimator could be generated in the form of a Kalman spectral density of the mirror displacement under this opti- filter for this application.However,a Kalman filter is con- mized modal damping control is plotted in Figure 5. cerned with optimal sensor data recovery whereas our goal is The need to include a model of the suspension within the to optimize the noise reaching the suspended mirror(stage 4). control algorithm suggests a limitation of the modal damping Sacrificing sensor signal accuracy and damping is acceptable approach.If the model is not accurate the inferred motion of to a certain degree if the noise performance is improved.Con- the lowest three stages may differ from reality.The dynamics sequently,a more direct approach is taken to obtain the values of the suspension depends on the positions of,for example, of Q and R. the attachment points relative to the masses and/or springs First,Q is set by placing the square of the inverse of the on each stage.In the aLIGO quadruple suspension,an error resonance frequencies on the diagonal.In this way lower res- of order 100 um in the vertical coordinate of an attachment onance frequencies,which have more mechanical energy.will point can affect the pitch modes significantly.This requires be damped more efficiently.Only the value of R remains to be the controller to be tuned to the suspension,at least in some optimized. If we restrict R to be diagonal,a common assumption, 10 we are left with six parameters to optimize simultaneously, relating to six Euler stage 1 measurement signals.To sim- 10 plify this problem one can invoke the symmetry of the pendu- 10 lum.For example,all the modes that represent vertical motion 10 (z)are decoupled from the others.Thus,these modes can be thought of as representing a separate system which has only 10 four DOFs and a single sensor signal.The same decompo- 10 sition can be made for yaw and x motion.The remaining 3 10 directions are inseparably coupled. settling cost,max(T) To choose the best value of R the control design is first 10 sensor noise cost,max(N total cost,Jg set so that the damping time requirement is met assuming full state information.An optimization routine in MATLAB then 10 10 10 R simulates the performance of the closed loop system for es- timators designed using many values of R over a sufficiently FIG.4.The components of the cost function Eq.(12)for the x DOF as a function of R calculated by the optimization routine.At each value of R the large space.A cost function,Eq.(12),dependent on the per- closed loop system performance is simulated using the estimator design based formance criteria is calculated for each value of R.The value on the LOR solution with that particular R value. Reuse of AlP Publishing content is subject to the terms at:https://publishing.aip.org/authors/nghts-and-perm D0wmlo8doP:183.195251.60:Fi.22Apr2016 00:5549044501-5 K. A. Strain and B. N. Shapiro Rev. Sci. Instrum. 83, 044501 (2012) The estimator equation in (8) receives the control input u and the stage 1 sensor signals y in the Euler frame and esti￾mates the corresponding modal signals ˆ q using the modeled state space matrices Am, Bm, Cm. The subscript m indicates the matrices refer to the modal frame. The goal is now to de￾sign the estimator feedback gain Lm. The observer separation principle allows us to keep the already established control scheme, provided that the estima￾tor has an accurate model of the pendulum, which is the as￾sumption here. Note that the estimator replaces the direct sig￾nal transformation in (5). The estimator is designed using the Linear Quadratic Regulator (LQR) technique which solves the cost function in Eqs. (9) and (10): J =  ∞ 0  [ ˜ qT ˙˜ qT ]Q ˜ q ˙˜ q + zT mRzm  dt, (9) with Lm = argmin Lm (J ). (10) The Q matrix weights the accuracy of the modal state es￾timation while the R matrix weights the cost of using a noisy measurement. Here zm is defined as zm = −LT m ˜ q ˙˜ q , (11) where Lm is the estimator feedback matrix determined by (10). An estimator could be generated in the form of a Kalman filter for this application. However, a Kalman filter is con￾cerned with optimal sensor data recovery whereas our goal is to optimize the noise reaching the suspended mirror (stage 4). Sacrificing sensor signal accuracy and damping is acceptable to a certain degree if the noise performance is improved. Con￾sequently, a more direct approach is taken to obtain the values of Q and R. First, Q is set by placing the square of the inverse of the resonance frequencies on the diagonal. In this way lower res￾onance frequencies, which have more mechanical energy, will be damped more efficiently. Only the value of R remains to be optimized. If we restrict R to be diagonal, a common assumption, we are left with six parameters to optimize simultaneously, relating to six Euler stage 1 measurement signals. To sim￾plify this problem one can invoke the symmetry of the pendu￾lum. For example, all the modes that represent vertical motion (z) are decoupled from the others. Thus, these modes can be thought of as representing a separate system which has only four DOFs and a single sensor signal. The same decompo￾sition can be made for yaw and x motion. The remaining 3 directions are inseparably coupled. To choose the best value of R the control design is first set so that the damping time requirement is met assuming full state information. An optimization routine in MATLABR then simulates the performance of the closed loop system for es￾timators designed using many values of R over a sufficiently large space. A cost function, Eq. (12), dependent on the per￾formance criteria is calculated for each value of R. The value that minimizes this cost function in the chosen interval is opti￾mal. For x, R is a scalar because it is a decoupled DOF by the symmetry of the pendulum. Coupled DOFs such as y, pitch, and roll must be considered simultaneously resulting in a ma￾trix. The two competing performance criteria of damping time and technical noise are stated above. The sensor noise is a known measured quantity approximately 7 × 10−11 m/ √Hz beyond 10 Hz. Since the contribution to mirror motion drops off quickly in the frequency domain, we will only consider the contribution at 10 Hz, the start of the gravitational wave detection band. These two performance criteria are represented in the op￾timization routine with the cost function: JR(R) = max DOF (T 2 s ) + max DOF (N2 ), (12) with R = argmin R (JR). (13) Here, Ts is the normalized stage 4 settling time; chosen as the maximum settling time of all the DOFs normalized by the goal, 10 s. Similarly, N is the maximum sensor noise con￾tribution to the mirror relative to the respective requirement at 10 Hz for each DOF considered. As an example, the results of Eq. (12) are plotted in Figure 4 for the x DOF where R is scalar valued. The opti￾mal value occurs at R = 0.06. Since the total cost is less than 1, both design requirements are met at this optimal point. The spectral density of the mirror displacement under this opti￾mized modal damping control is plotted in Figure 5. The need to include a model of the suspension within the control algorithm suggests a limitation of the modal damping approach. If the model is not accurate the inferred motion of the lowest three stages may differ from reality. The dynamics of the suspension depends on the positions of, for example, the attachment points relative to the masses and/or springs on each stage. In the aLIGO quadruple suspension, an error of order 100μm in the vertical coordinate of an attachment point can affect the pitch modes significantly. This requires the controller to be tuned to the suspension, at least in some FIG. 4. The components of the cost function Eq. (12) for the x DOF as a function of R calculated by the optimization routine. At each value of R the closed loop system performance is simulated using the estimator design based on the LQR solution with that particular R value. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 183.195.251.6 On: Fri, 22 Apr 2016 00:55:49